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Question
Differentiate the following with respect to x.
`(x^2 + x + 1)/(x^2 - x + 1)`
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Solution
Let y = `(x^2 + x + 1)/(x^2 - x + 1)`
`"d"/"dx" ("u"/"v") = ("v" "du"/"dx" - "u" "du"/"dx")/"v"^2`
`"dy"/"dx" = ((x^2 - x + 1)"d"/"dx" (x^2 + x + 1) - (x^2 + x + 1) "d"/"dx" (x^2 - x + 1))/(x^2 - x + 1)^2`
`= ((x^2 - x + 1)(2x + 1) - (x^2 + x + 1)(2x - 1))/(x^2 - x + 1)^2`
`= (2x^3 - 2x^3 + x^2 - 2x^2 - 2x^2 + x^2 + 2x - x - 2x + x + 1 + 1)/(x^2 - x + 1)^2`
`= (- 2x^2 + 2)/(x^2 - x + 1)^2`
`= (- 2(x^2 - 1))/(x^2 - x + 1)^2` (or)
`= (2 (1 - x^2))/(x^2 - x + 1)^2`
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